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Kähler–Einstein metrics with SU(2) action

Published online by Cambridge University Press:  24 October 2008

Andrew S. Dancer  and
Ian A. B. Strachan
Andrew S. Dancer
Affiliation:
Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, D53225 Bonn 3, Germany
Ian A. B. Strachan
Affiliation:
Department of Mathematics and Statistics, University of Newcastle, Newcastle-on-Tyne, NEl 7RU
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Extract

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined by

is closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 3 , May 1994 , pp. 513 - 525
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Atiyah, M. F. and Hitchin, N. J.. The geometry and dynamics of magnetic monopoles (Princeton University Press, 1988).Google Scholar
[2]Belinskii, V. A., Gibbons, G. W.. Page, D. N. and Pope, C. N.. Asymptotically Euclidean Bianchi IX metrics in quantum gravity. Phys. Lett 76B (1978), 433–435.Google Scholar
[3]Bergery, L. Berard. Sur des nouvelles variétés riemanniennes d'Einstein. Publications de l'Institut E. Cartan (Nancy) 4 (1982), 1–60.Google Scholar
[4]Besse, A.. Einsteina Manifolds. Ergebnisse der Math. und ihrer Grenzgebiete (Springer-Verlag, 1987).Google Scholar
[5]Bouchiat, C. and Gibbons, G. W.. Non-integrable quantum phase in the evolution of a spin-1 system: a physical consequence of the non-trivial topology of the quantum state-space. J. Phys. France 49 (1988), 187–199.Google Scholar
[6]Futaki, A.. Kahler–Einstein metrics and integral invariants. Springer Lecture Notes in Mathematics 1314 (Springer-Verlag, 1988).Google Scholar
[7]Gibbons, C. V. and Pope, C. N.. The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys. 66 (1979), 267–290.Google Scholar
[8]Jensen, G. R.. Homogeneous Einstein spaces of dimension 4. J. Differential Geometry 3 (1969), 309–349.Google Scholar
[9]Matsushima, Y.. Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne. Nagoya Math J. 11(1957)145–150.Google Scholar
[10]Myers, S. B.. Riemannian geometry in the large. Duke Math. J. 1 (1935), 39–49.CrossRefGoogle Scholar
[11]Pedersen, H.. Eguchi–Hanson metrics with cosmological constant. Classical and Quantum Gravity 2 (1985), 579–587.Google Scholar
[12]Pedersen, H. and Poon, Y. S.. Kähler surfaces with zero scalar curvature. Classical and Quantum Gravity 7 (1990), 1707–1719.Google Scholar